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--- docs:classes:flowfield [2009/02/16 11:44]
gibson
+++ docs:classes:flowfield [2010/02/02 07:55]
@@ Line 1: / Line 1: @@
-====== FlowField ======
-The FlowField class represents vector-valued fields of the form
-<latex>
-{\bf u}({\bf x}) &= \sum_{k_x,k_y,k_z,j}  \hat{u}_{k_x k_y k_z j}\bar{T}_{k_y}(y) \; e^{2 \pi i (k_x x/L_x + k_z z/L_z)} {\bf e}_j
-</latex>
-and also scalar and tensor fields with appropriate changes in the dimensionality of the coefficients.
-The barred %%T%% function is a Chebyshev polynomial scaled to fit the domain y ∈ [a,b]. ((<latex>
-\bar{T}_{k_y}(y) = T_{k_y}\left(\frac{2}{b-a}(y - \frac{b+a}{2})\right)</latex>)) The spatial domain
-of a FlowField is  Ω = [0,Lx] x [a,b] x [0,Lz], with periodicity in x and z.
-In channelflow programming, fields such as velocity, pressure, stress tensors, vorticity, etc.
-are stored as variables of type FlowField. The main functionality of the FlowField class is
-   * algebraic and differential operations, +/-, +=, ∇, ∇<sup>2</sup>, norms, inner products, etc.
-   * transforming back and forth between spectral coefficients <latex> \hat{u}_{k_x k_y k_z j}</latex> and gridpoint values <latex>u_j (x_{n_x}, y_{n_y}, z_{n_z})</latex>
-   * serving as input to DNS algorithms, which map velocity fields forward in time: u(x,t) → u(x, t+Δt)
-   * setting and accessing scpetral coefficients and gridpoint values
-   * reading and writing to disk
-===== Constructors / Initialization =====
-FlowFields are initialized with gridsize and cellsize parameters, read from disk,
-or assigned from computations. Examples:
-<code c++>
-   FlowField f;                                   // null value, 0-d field on 0x0x0 grid
-   FlowField u(Nx, Ny, Nz, Nd, Lx, Lz, a, b);     // Nd-dim field on Nx x Ny x Nz grid, [0,Lx]x[a,b]x[0,Lz]
-   FlowField g(Nx, Ny, Nz, Nd, 2, Lx, Lz, a, b);  // Nd-dim 2-tensor
-   FlowField h("h");                              // read from file "h.ff"
-   FlowField omega = curl(u);
-</code>
-===== Algebraic and differential operators =====
-Assume f,g,h etc. are FlowField variables with compatible cell and grid sizes. Examples of
-possible operations
-<code>
-   f += g;                  // f = f + g
-   f = curl(g);
-   f = lapl(g);
-   f = div(g);
-   f = diff(g, j, n);       // f_i  = d^n g_i /dx_j
-   f = grad(g);             // f_ij = dg_i / dx_j
-   f = cross(g,h);
-   f *= 2.7;                // f = 2.7*f
-   Real c = L2IP(f,g);      // L2 inner product of f,g
-   Real n = L2Norm(u);
-   Real D = dissipation(u);
-   Real E = energy(u);
-   Real I = wallshear(u);
-</code>
-The latter functions are defined as
-<latex> $ \begin{align*}
-L2IP(f,g) &= \frac{1}{L_x L_y L_z} \int_{\Omega} {\bf f} \cdot {\bf g} \,\, d{\bf x} \\
-L2Norm(u) &= \left(\frac{1}{L_x L_y L_z} \int_{\Omega} \|{\bf u}\|^2\,\, d{\bf x} \right)^{1/2}\\
-     E(u) &= \frac{1}{2 L_x L_y L_z} \int_{\Omega} \|{\bf u}\|^2\,\, d{\bf x}  \\
-     D(u) &= \frac{1}{L_x L_y L_z} \int_{\Omega} \|\nabla \times {\bf u}\|^2 \,\, d{\bf x} \\
-     I(u) &= \frac{1}{L_x L_z} \int_{y=a,b} \frac{\partial u}{\partial y} \, dx dz
-\end{align*} $ </latex>
-===== Transforms and data access =====
-FlowField transforms are a complicated subject --there are transforms in x,y, and z and
-implicit symmetries in complex spectral coefficients due to the real-valuedness of the
-field, for instance. This section outlines the bare essentials of transforms and data
-access methods. For further details see the {docs:chflowguide.pdf|Channelflow User Guide}}.

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